Question

The differential equation of the family of curve $y=A{e}^{3x}+B{e}^{5x}$, where A, B are arbitrary constants is :

• A. $\frac{d^{2}y}{d{x}^2}+8\frac{dy}{dx}=15y=0$
• B. $\frac{d^{2}y}{d{x}^2}-8\frac{dy}{dx}+15y=0$
• C. $\frac{d^{2}y}{d{x}^2}-\frac{dy}{dx}=y=0$
• D. None of these

Answer 1

Answer: (B)

$y=A{e}^{3x}+B{e}^{5x}$
$\Rightarrow \frac{dy}{dx}=3A{e}^{3x}+5B{e}^{5x}\dots \left(1\right)$
$\Rightarrow \frac{d^{3}y}{d{x}^2}=9A{e}^{3x}+25B{e}^{5x}\dots \left(2\right)$
$\therefore \frac{d^{2}y}{d{x}^2}-8\frac{dy}{dx}+15y$
$=9A{e}^{3x}+25B{e}^{5x}-24A{e}^{3x}-40B{e}^{5x}$
$+15A{e}^{3x}+15B{e}^{5x}=0$